On abelian automorphism groups of hypersurfaces

Abstract

Given integers d ≥ 3 and N ≥ 3, let G be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree d in the complex projective space ℙN−1. Suppose G ⊂ PGL(N, ℂ) can be lifted to a subgroup of GL(N, ℂ). Suppose moreover that there exists an element g in G such that G/❬g❭ has order coprime to d − 1. Then all possible G are determined (Theorem 4.3). As an application, we derive (Theorem 4.8) all possible orders of linear automorphisms of smooth hypersurfaces for any given (d, N). In particular, we show (Proposition 5.1) that the order of an automorphism of a smooth cubic fourfold is a factor of 21, 30, 32, 33, 36 or 48, and each of those 6 numbers is achieved by a unique (up to isomorphism) cubic fourfold.